When applying GRAPE to find the time-optimal pulse for a C\(_2\) Z gate in the triangular arrangement at a finite \(B\) , pulses with a duration significantly below the duration of the time-optimal pulse in the \(B=\infty \)  case are found. As an example, at \(B=10\Omega _{\mathrm {max}}\)  there exists a pulse with duration \(T\Omega _{\mathrm {max}}= 10\)  and gate error \(1-F = 3\cdot 10^{-10}\) , which is shown in Fig. REF (a). The phase \(\varphi (t)\)  of this pulse varies rapidly with \(t\) , such that the detuning \(\Delta = \mathrm {d}\varphi / \mathrm {d}t\)  is of the order of \(B\)  [Fig. REF (b)]. This suggests that this pulse operates in the so-called Rydberg antiblockade regime [1]} where states with two atoms in the Rydberg state can become significantly populated. This is confirmed by the populations of \(\mathinner {|{0rr}\rangle }\)  and \(\mathinner {|{1rr}\rangle }\) , that are shown as a function of \(t\)  in Fig.REF (c) as blue solid line and orange dashed line, respectively. The pulse therefore speeds up the C\(_2\) Z gate by using states with two atoms in the Rydberg state, which is not possible in the \(B=\infty \)  case. Naturally one would expect such a pulse to be very sensitive to variations of \(B\) , and indeed when decreasing \(B\)  by just 10% the gate error increases drastically to \(1-F = 0.59\) . At a finite \(B\)  it is therefore possible to significantly speed up the C\(_2\) Z gate at the cost of a high sensitivity to variations in the blockade strength. Since the blockade strength depends strongly on the distance of the atoms, this pulse is only feasible if the distance between the atoms is known with high precision.
